Chapter 10: Problem 28
Find the equation of the given central conic. Hyperbola with foci \((\pm 4,0)\) and directrices \(x=\pm 1\)
Short Answer
Expert verified
The hyperbola's equation is \(x^2 - \frac{y^2}{15} = 1\).
Step by step solution
01
Identify the center and orientation of the hyperbola
Since the foci are located at \((\pm 4,0)\), the center of the hyperbola is at the origin \((0,0)\). The given foci positions indicate that the transverse axis is horizontal.
02
Determine the relationship between constants of the hyperbola
For a hyperbola, the relationship between the semi-major axis \(a\), the distance to the foci \(c\), and the semi-minor axis \(b\), is given by the equation \(c^2 = a^2 + b^2\). The foci \((\pm 4, 0)\) provide \(c = 4\).
03
Use directrix information to find \(a\)
The equation involving the directrices is \(\frac{c}{a} = e\), where \(e\) is the eccentricity and equals \(\frac{c}{a}\). The directrix equation \(x = \pm 1\) leads to \(c/a = 4/a = 4\), giving \(a = 1\).
04
Calculate \(b\) using conic relationship
Substitute \(a = 1\) and \(c = 4\) into the relationship \(c^2 = a^2 + b^2\): \[16 = 1 + b^2\] Thus, \(b^2 = 15\) and \(b = \sqrt{15}\).
05
Write the standard equation of the hyperbola
With the values determined, substitute into the standard horizontal hyperbola equation:\[\frac{x^2}{1^2} - \frac{y^2}{15} = 1\]This simplifies to:\[x^2 - \frac{y^2}{15} = 1\]
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Key Concepts
These are the key concepts you need to understand to accurately answer the question.
Conic Sections
Conic sections are a fascinating part of geometry that describe the different shapes we can obtain by slicing a cone with a plane. These include circles, ellipses, parabolas, and hyperbolas. Each shape has unique characteristics, but all fall under the umbrella of conic sections because of their geometric origins.
When we specifically talk about hyperbolas, they are the conic sections that arise when the slicing plane intersects both halves of the cone, typically resulting in two separate curves known as branches. A hyperbola can be thought of as an "open" ellipse. Unlike ellipses, hyperbolas have two axes and two separate, symmetrical branches.
The defining property of a hyperbola includes its two foci (focal points). For any point on a hyperbola, the absolute difference in distances to these two foci is constant. This unique property differentiates hyperbolas from other conic sections. Hyperbolas also feature asymptotes—lines that the branches approach indefinitely. The orientation of the hyperbola depends on the positioning of the foci: horizontal if the foci are aligned along the x-axis, and vertical if along the y-axis.
When we specifically talk about hyperbolas, they are the conic sections that arise when the slicing plane intersects both halves of the cone, typically resulting in two separate curves known as branches. A hyperbola can be thought of as an "open" ellipse. Unlike ellipses, hyperbolas have two axes and two separate, symmetrical branches.
The defining property of a hyperbola includes its two foci (focal points). For any point on a hyperbola, the absolute difference in distances to these two foci is constant. This unique property differentiates hyperbolas from other conic sections. Hyperbolas also feature asymptotes—lines that the branches approach indefinitely. The orientation of the hyperbola depends on the positioning of the foci: horizontal if the foci are aligned along the x-axis, and vertical if along the y-axis.
Eccentricity
Eccentricity (\(e\)) is a critical concept when studying conic sections, and especially hyperbolas. It measures how "stretched" a conic section is compared to a circle. For hyperbolas, the eccentricity is always greater than 1, which reflects their open, wide shape.
The formula for the eccentricity of a hyperbola is \(e = \frac{c}{a}\), where \(c\) is the distance from the center to a focus, and \(a\) is the distance from the center to the vertices along the transverse axis. The directrix, a line perpendicular to the transverse axis, further helps define how much the conic section deviates from being circular.
In the exercise, the hyperbola has foci at \((\pm 4, 0)\) and directrices at \(x = \pm 1\). Using this, we determine that \(e\) is 4, since the formula \(\frac{c}{a} = e\) gives \(4\) (foci distance over vertex distance). This high eccentricity indicates the hyperbola is highly open compared to circular shapes like circles or ellipses.
The formula for the eccentricity of a hyperbola is \(e = \frac{c}{a}\), where \(c\) is the distance from the center to a focus, and \(a\) is the distance from the center to the vertices along the transverse axis. The directrix, a line perpendicular to the transverse axis, further helps define how much the conic section deviates from being circular.
In the exercise, the hyperbola has foci at \((\pm 4, 0)\) and directrices at \(x = \pm 1\). Using this, we determine that \(e\) is 4, since the formula \(\frac{c}{a} = e\) gives \(4\) (foci distance over vertex distance). This high eccentricity indicates the hyperbola is highly open compared to circular shapes like circles or ellipses.
Coordinate Geometry
Coordinate geometry, sometimes referred to as analytic geometry, is the study of geometry using a coordinate system. This allows us to analyze shapes and structures in a plane with algebraic equations, offering powerful techniques to solve diverse geometric problems.
With hyperbolas, coordinate geometry facilitates a deeper understanding through its standard equation form, which helps visualize the curve's structure: \(\frac{x^2}{a^2} - \frac{y^2}{b^2} = 1\). By substituting values into this form, we can understand the orientation and dimensions of the hyperbola. For instance, this exercise determines values of \(a = 1\) and \(b = \sqrt{15}\), using the relationship \(c^2 = a^2 + b^2\).
By solving for \(b\), we find the relationship between the axes of our hyperbola. This interaction of algebra and geometry makes it easier to derive the precise equation of a specific conic section such as the hyperbola in this exercise, thus allowing us to visualize and graph complex curves easily.
With hyperbolas, coordinate geometry facilitates a deeper understanding through its standard equation form, which helps visualize the curve's structure: \(\frac{x^2}{a^2} - \frac{y^2}{b^2} = 1\). By substituting values into this form, we can understand the orientation and dimensions of the hyperbola. For instance, this exercise determines values of \(a = 1\) and \(b = \sqrt{15}\), using the relationship \(c^2 = a^2 + b^2\).
By solving for \(b\), we find the relationship between the axes of our hyperbola. This interaction of algebra and geometry makes it easier to derive the precise equation of a specific conic section such as the hyperbola in this exercise, thus allowing us to visualize and graph complex curves easily.
